Question

One-fourth of the residents of the Burning Ridge Estates leave their garage doors open when they are away from home. The local chief of police estimates that 5 % of the garages with open doors will have something stolen, but only 1 percent of those closed will have something stolen. If the garage is robbed, what is the probability the doors ware left open

Answer 1

If the garage is robbed, the probability the doors ware left open is 0.625.

Explanation:

Denote the events as follows:

X = garage doors open

Y = something is stolen

The information provided is:

P (X) = 0.25

P (Y|X) = 0.05

P (Y|X') = 0.01

Compute probability that the garage is robbed as follows:

`P(Y)=P(Y|X)P(X)+P(Y|X')P(X')\\\\=(0.05* 0.25)+(0.01* (1-0.25))\\\\=0.0125+0.0075\\\\=0.02`

Compute the probability the doors ware left open given that the garage is robbed as follows:

`P(X|Y)=(P(Y|X)P(X))/(P(Y))\\\\=(0.05* 0.25)/(0.02)\\\\=0.625`

Thus, if the garage is robbed, the probability the doors ware left open is 0.625.